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Doctoral Thesis
DOI
10.11606/T.43.2000.tde-28112013-102436
Document
Author
Full name
Júlio César Bastos de Figueiredo
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2000
Supervisor
Committee
Malta, Coraci Pereira (President)
Caldas, Ibere Luiz
Furuie, Sergio Shiguemi
Koiller, Jair
Ranvaud, Ronald Dennis Paul Kenneth Clive
Title in Portuguese
Equações Diferenciais não Lineares com Três Retardos: Estudo Detalhado das Soluções
Keywords in Portuguese
Equações diferenciais não lineares
Física teórica
Abstract in Portuguese
In this thesis we study the behavior of a simple control system based on a delay differential equation with multiple loops of negative feedback. Numerical solutions of the delay differential equation with N delays d/dt x(t) = -x(t) + 1/N POT.N IND.i=1 / POT.n IND.i + x (t- IND.i) POT.n have been investigated as function of its parameters: n, i and i. A simple numerical method for determine the stability regions of the equilibrium points in the parameter space (i, n) is presented. The existence of a doubling period route to chaos in the equation, for N = 3, is characterized by the construction of bifurcation diagram with parameter n. A numerical method that uses the analysis of Poincaré sections of the reconstructed attractor to find aperiodic solutions in the parameter space of the equation is also presented. We apply this method for N = 2 and get evidences for the existence of chaotic solutions as result of a period doubling route to chaos (chaotic solutions for N = 2 in that equation had never been observed). Finally, we study the solutions of a piecewise constant equation that corresponds to the limit case n .
Title in English
Nonlinear differential equations with three delays: detailed study of the solutions.
Keywords in English
Nonlinear differential equations
Theoretical physics
Abstract in English
In this thesis we study the behavior of a simple control system based on a delay differential equation with multiple loops of negative feedback. Numerical solutions of the delay differential equation with N delays d/dt x(t) = -x(t) + 1/N POT.N IND.i=1 / POT.n IND.i + x (t- IND.i) POT.n have been investigated as function of its parameters: n, i and i. A simple numerical method for determine the stability regions of the equilibrium points in the parameter space (i, n) is presented. The existence of a doubling period route to chaos in the equation, for N = 3, is characterized by the construction of bifurcation diagram with parameter n. A numerical method that uses the analysis of Poincaré sections of the reconstructed attractor to find aperiodic solutions in the parameter space of the equation is also presented. We apply this method for N = 2 and get evidences for the existence of chaotic solutions as result of a period doubling route to chaos (chaotic solutions for N = 2 in that equation had never been observed). Finally, we study the solutions of a piecewise constant equation that corresponds to the limit case n .
 
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30820Figueiredo.pdf (1.54 Mbytes)
Publishing Date
2013-12-18
 
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